A Set of Second-Order Differential Equations Associated with 



Reflections in Rectangular Wave Guides — Application 



to Guide Connected to Horn* 



By S. O. RICE 



In dealing with corners and similar irregularities in rectangular wave guides 

 it is sometimes helpful to transform the system, conformally, into a straight 

 guide. Propagation in the straight guide may then be studied by an integral 

 equation method, as is done in a companion paper, or by a more general method 

 based upon a certain set of ordinary differential equations. Here the second 

 method is developed and applied to determine the reflection produced at the junc- 

 tion of a straight guide and a sectoral horn — a problem the first method is unable 

 to handle. The WKB approximation for a single second-order differential 

 equation is extended to a set of equations and approximate expressions for the 

 reflection coefhcient are derived. 



IN A companion paper^ the disturbance produced by a corner in a rec- 

 tangular wave guide is examined by transforming the system, con- 

 formally, into a straight guide. Although the medium in the straight guide 

 is no longer uniform, an integral equation may be set up and approximate 

 solutions obtained. 



In that paper the wave guide is assumed to have the same cross-section 

 at -f CO as at — oc . WTien this is not so, a conformal transformation may 

 still be used to transform the system into a straight guide provided one 

 dimension of the original cross-section is constant. However, now some 

 advantage appears to be gained by replacing the integral equation by a set 

 of differential equations. Since two cases appear, corresponding to E and // 

 corners, there are two sets of equations to be considered. 



These two sets of equations are studied in the present paper. After their 

 derivation in Sections 1 and 2 several remarks are made in Section 3 con- 

 cerning their solution, special emphasis being laid on the problem of deter- 

 mining the reflection coefficient. In the remainder of the paper the general 

 theory is applied to a system formed by joining a rectangular w^ave guide 

 to a horn (with plane sides) flared in one direction. The reflection coeffi- 

 cients for sectoral horns flared in the planes of the electric and magnetic 

 intensity, respectively, are given approximately by equations (6-1) and (7-1). 

 These approximations assume the angle of flare to be small so that, as it 

 turns out, only the first equations of the respective sets need be considered. 



As was mentioned in the companion paper, Robert Piloty has recently 

 made use of conformal transformations in wave guide problems. In his 



* Presented at the Second Symposium on AppUed Mathematics, Cambridge, Mass., 

 July 29, 1948. 



'See list of references at end of paper. 



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